leontif input output

Hi, um i just failed my linear exam so but I'm going to take a supplementary exam in a few days so i was wondering if someone could please help me. Because my lecturer is on holiday as its uni holidays now.

Well this is a leontif input-output question. This question comes from my lecture notes, where the model leads to the linear system,

and it says that the answer is(to the nearest decimal point):

But when i did it i got a totally different answer and then decided to check the online calculator:

I'm just wondering if someone could tell me if my lecture notes are wrong? as i have no idea if they're correct or not. Coz i thought that for leontif input output, all you have to do is get it into reduced row echelon form?

I'd solve the problem exactly as in your link (Linear Algebra Toolkit). The method is good. I won't check all the details to see if the answer is good (afterall you can plug the solution into the first 3 equations to see if they are satisfied).
So for your exam, if they ask to find the solutions of a system of linear equations, you can form an augmented matrix and row reduce it.